COMPLETE MONOTONICITY INVOLVING THE DIVIDED DIFFERENCE OF POLYGAMMA FUNCTIONS

For r, s is an element of R and rho= min {r, s}, let D [x r, x s; psi n-1] equivalent to -phi n(x) be the divided difference of the functions psi n-1 = (-1)n psi(n-1) (n is an element of N) on (-rho, infinity), where psi(n) stands for the polygamma functions. In this paper, we present the necessary and sufficient conditions for the functions x 7 -> x 7 -> pi k i=1 pi k i=1 phi mi (x) - lambda k pi k i=1 phi ni (x) , phi ni (x) - mu k phi snk (x) to be completely monotonic on (-rho, infinity), where mi, ni is an element of N for i = 1, .., k with k >= 2 and snk = sigma ki=1 ni. These generalize known results and gives an answer to a problem.